Linear Model Adjustment and Approximate Approach for Creating

Minimal Overhead Wires Network for Vehicle Schedules

Dobroslav Grygar

and Michal Kohni

Department of Mathematical Methods and Operations Research, Faculty of Management Science and Informatics,

University of Zilina, Univerzitna 8215/1, 010 26 Zilina, Slovakia

Keywords:

Battery Assisted Trolleybus, Exact Approach, Mathematical Model, Optimization, Schedule, Route.

Abstract:

Nowadays, there is a signiﬁcant effort in reducing the environmental impact caused by public transport. This

goal can be achieved in many ways. One possible way is in using battery-assisted trolleybuses in cities.

Therefore this paper deals with the question of how to create a minimum overhead contact network for such

vehicles operation. The article presents the mathematical model of such a problem and tests the impact of

the version with the modiﬁed condition. It also proposes a suboptimal way to propose needed vire network

for operation selected vehicle schedules using individual routes. As benchmarks, some vehicle schedules in

Zilina were selected.

1 INTRODUCTION

The European Union is currently calling for the

reduction of the environmental impact caused by

carbon-dioxide emissions. It can be assumed that

the trend of implementation of battery-assisted trol-

leybuses in cities will grow soon. The technology

of battery-assisted trolleybus combines the beneﬁts of

classical trolleybus with movement freedom of stan-

dard buses. The dasic idea is illustrated in Figure 1.

Figure 1: The basic idea of battery assisted trolleybus oper-

ation. [Vossloh Kiepe] (Bartomiejczyk, 2017).

We have developed the mathematical model to

create a minimal network of overhead wires that

would be sufﬁcient for deployment and operation of

https://orcid.org/0000-0002-5834-6624

https://orcid.org/0000-0002-9421-4899

such battery-assisted trolleybuses for individual bus

lines (Grygar and Kohani, 2019b). Next logical step

in our research is to use this model to solve the prob-

lem with longer and more complex vehicle schedules.

This paper introduces the mathematical model of ex-

act optimization for individual routes. We are also

dealing with the question, how to improve the effec-

tivity of our mathematical model. Therefore, we will

compare the adjusted version of the linear model with

the original formulation.

With this model, we are able to optimize all of the

service trips in the vehicle schedule individually. Us-

ing correct parameters, we ensure that a vehicle will

be able to complete all tours in schedule (Grygar and

Kohani, 2019a).

This approach can be considered as a heuristic al-

gorithm extension to the previous research. Based on

this approach, it is possible to optimize the schedule

of a vehicle without regard to its length. Although

obtained results are based on an exact approach to op-

timizing the individual routes, it is necessary to con-

sider the coverage of individual schedules as a heuris-

tic result. We are using real data from the public trans-

port system. The data was obtained from the public

transport provider (DPMZ) in Zilina (Grygar and Ko-

hani, 2019a).

This paper is written with the following structure.

First, we mention the current state of the art in Section

2. Then, in the next section of the paper, we formulate

the problem. After that, Section 4 describes the data

Grygar, D. and Kohni, M.

Linear Model Adjustment and Approximate Approach for Creating Minimal Overhead Wires Network for Vehicle Schedules.

DOI: 10.5220/0008966501870193

In Proceedings of the 9th International Conference on Operations Research and Enterprise Systems (ICORES 2020), pages 187-193

ISBN: 978-989-758-396-4; ISSN: 2184-4372

187

conversion approach. Section 5 explains the mathe-

matical model of the problem and studied condition

modiﬁcation. Lastly, Section 6 introduces performed

benchmarks and explains the collected results.

2 STATE OF THE ART

Traditional trolleybuses have been used for several

years in public transport in some cities in Slovakia. To

deploy trolleybuses, it is necessary to cover all routes

with continuous overhead wires (Pisko, 2017). An

interesting alternative is in using battery-assisted trol-

leybuses.

The battery-assisted trolleybuses are considered

as the most cost-effective electrically powered pub-

lic transport technology solution (Bergk et al., 2016),

(Rogge et al., 2015) and (Yang et al., 2014).

In our previous research, we published the analy-

sis in the topic of limiting factors of battery-assisted

trolleybuses (Grygar et al., 2019). This study is

based on valuable information papers from other au-

thors. Vehicle technology is well cowered in real de-

ployment observations by (Bartomiejczyk, 2017) and

(Bartomiejczyk et al., 2013). Information about lifes-

pan of traction batteries and temperature impact can

be found in (Ghlich et al., 2014), (Montoya et al.,

2017), (Rice et al., 2017), (Ritter et al., 2016), (Rogge

et al., 2015) and (Saft, 2016).

Unfortunately, there are no studies available to ad-

dress the problem of building an optimal overhead

wires network for battery-assisted trolleybuses in a

big graph. We can get inspiration from articles deal-

ing with the deployment of induction lines for pas-

senger cars. This task is considered similar (He

et al., 2016), (Hwang et al., 2018) and (Ushijima-

Mwesigwa et al., 2017). Our proposed model is

based on a modiﬁcation of the model presented in

(Ushijima-Mwesigwa et al., 2017).

There are published many works about scheduling

electric vehicles in public transport. For example pa-

per (Kooten Niekerk et al., 2017) extends traditional

scheduling problem by using electric vehicles. Fur-

thermore, there is better paper (Janovec and Kohni,

2019) also deals with this topic. Authors use more

realistic linear model and it is tested with real word

benchmarks from Zilina, Slovakia.

Our problem can be basically explained as lo-

cation problem on edges. There are many papers

dealing with all sorts of standard location problems

(Jaosikova and Jankovic, 2018), (Janosikova et al.,

2017), (Kvet, 2019) or (Janacek and Kvet, 2016). We

may explore options on how to use a similar approach

to our problem in future research.

3 THE PROBLEM

FORMULATION

In our research, we often deal with the problem of

designing network systems, such as rescue or distri-

bution systems. In such tasks, we can meet with an

unlimited or limited capacity problem. When solving

a standard location problem, we are usually selecting

locations for centres in network nodes (Janek and Ko-

vaikov, 1999). These can be tasks oriented to placing

shops, warehouses, stations, hospitals and so on. The

centres capacity or distribution times are represented

as model conditions. Objective function is often ori-

ented to minimizing the total cost (Janek et al., 2015).

Our problem can be formulated as a location prob-

lem on edges. So we are trying to create a minimal

network of overhead wires that would be enough for

deployment and operation of battery-assisted trolley-

buses technology. The building cost of required over-

head contact wires represents a big part of the cost of

the entire transport system. Therefore, we are about

to minimize it (Grygar et al., 2019).

In the current state of our research, we have to ex-

plore options on how to cover whole vehicle sched-

ules with overhead contact lines using current model

formulation. So, we introduce the suboptimal ap-

proach to optimize selected vehicle schedules. From a

computational point of view, the proposed mathemat-

ical model is inadequate for solving large scale tasks

with commercial solver like XPRESS IP. The options

of efﬁcient conditions modiﬁcations need to be also

explored.

The data conversion process is an important part

of solving the formulated problem. After data con-

version, we can start the process of ﬁnding exact so-

lutions for different datasets using selected IP solver.

4 DATA CONVERSION

To prepare data for solving using formulated model,

we need to perform the transformation of the road

network to another form. The process converts the

road network graph to road segment graph for each

route. In this transformation, we also convert edges

form road segment graph to nodes in the road seg-

ment graph and vice versa. This transformation is il-

lustrated in Figure 2 (Grygar and Kohani, 2019b).

State of Charge (SOC) represents actual state en-

ergy in a vehicles battery pack in %. Battery assisted

trolleybus starts a journey with an initial SOC. SOC of

a vehicle is calculated depending on whether a charg-

ing wire is built on the road segment (Grygar and Ko-

hani, 2019b).

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

188

Figure 2: Creation of road segments graph from road net-

work graph. a) Example of the road network graph. Bus

line (with stops 1, 3, 7, 6) is marked with the orange line.

Nodes are bus stops and edges are road segments. b) road

segment graph created for the mentioned bus line. Nodes

are road segments, AN is artiﬁcial node representing SOC

of a vehicle after compleating the trip (Grygar and Kohani,

2019b).

In the next phase, the state of charge graph is cre-

ated for each line. Vehicle SOC graph illustration is

in Figure 3. The mentioned graph represents all pos-

sible combinations of vehicles SOC. These options

are combinations of individual segments coverage of

a bus line by overhead charging wires. SOC graph

contains only these combinations where the vehicle

can pass whole route (Grygar and Kohani, 2019b).

Figure 3: State of charge graph. Rows represent individ-

ual charge level and nodes are road segments. Transitions

between charge level are marked with 1 of the vehicle is

charged from wires and with 0 if not. The y

(

s) and x

(

r, i)

are decision variables, and w

(

r, s, i) input data (individual

feasible combinations of charging). (Grygar and Kohani,

2019b).

5 MATHEMATICAL MODEL

FORMULATION

In this section, we describe the original version of the

linear model of our problem. Then we will describe

the proposed adjustment of the model.

We deﬁne set of all routes R, and set I

as feasi-

ble alternatives for route r and then set off all used

segments S. The number of segments n and routes m.

After that x

r,i

∈ {0, 1} is decision variable of selec-

tion of alternative i for route r. And is decision vari-

able y

∈ {0, 1} of segment s in graph will or won’t be

covered by overhead charging wire.

Input combinations w

r,s,i

∈ {0, 1} for route r on

segment s for alternative i is charging line needed.

Lastly the number of feasible alternatives a

for route

r and cost of building charging wires c

for segment

min

∑

s=1

· y

, (1)

a(r)

∑

i=1

r,i

= 1 f or r = 1..m, (2)

n · y

≥

∑

r=1

a(r)

∑

i=1

r,i

· w

r,s,i

f or s = 1..n, (3)

∈ {0, 1} f or s = 1..n, (4)

r,i

∈ {0, 1} f or r = 1..m; i = 1..a(r). (5)

The objective function 1 for the problem of mini-

mizing the total building cost. Constraint 2 serve for

choosing one combination for all routes on the line.

Constraint 3 ensures that we install a charging line if

at least one route requires an installation. The obliga-

tory constraints are 4 and 5.

5.1 Edited Mathematical Model

As we have presented in the already published re-

search, this formulation of the mathematical model is

not suitable for larger-scale tasks. Therefore, we de-

cided to make a change to the condition 3. Original

conditions 3 were formulated for each route.

Modiﬁcation of the condition 3 results in the fol-

lowing condition 6.

≥

a(r)

∑

i=1

r,i

· w

r,s,i

f or s = 1..n; r = 1..m. (6)

This change does not change the meaning of the

model. Conditions 6 are formulated for each service

trip and also for each route. It results in an increase

in the number of conditions when solving a task using

IP-solver XPRESS. We will study the impact of this

change on the solving process of the problem using

the following computational tests.

Linear Model Adjustment and Approximate Approach for Creating Minimal Overhead Wires Network for Vehicle Schedules

189

6 THE BENCHMARKS AND

COMPUTATIONAL TESTS

We obtained data from the public transport company

(DPMZ) in Zilina, Slovakia. Two vehicle sched-

ules were selected. For each benchmark, two-vehicle

schedules were selected.

Our model is able to solve the problem for sin-

gle bus routes only. Therefore, it was necessary to

ensure that the vehicle starts and ends individual jour-

neys on the line with the same level of SOC. There-

fore, we choose the rule that every service trip must

end with SOC of at least 60 %, and every service trip

start’s with SOC of 60 %. The solution found using

this parameter setting will mean that a vehicle is able

to overcome the entire schedule. However, it is not

guaranteed to ﬁnd the optimal solution to the original

problem. All input parameters related to vehicle are

listed in Table 1.

As it was mentioned before, we also studied the

impact of model adjustment. We tested both models

using the same benchmarks.

Table 1: Battery assisted trolleybus parameters settings se-

lected for all benchmark runs.

Parameter Value Unit

Min SOC 20 %

Max SOC 80 %

Battery capacity 50 kWh

Energy consuption 1.5 kWh/km

Initial SOC 60 %

Minimal SOC on route end 60 %

Charging time 30 s/1kWh

Travel speed 30 km/h

We used workstation with the following hard-

ware speciﬁcations. Processor Intel Core i5-7200U

2,5Ghz with 3,10Ghz turbo boost (two cores and

four threads), paired with 16 GB of DDR4 2133MHz

RAM. We selected IP-solver XPRESS IVE as soft-

ware tool for solving this problem.

6.1 Benchmark 4 4a and 5 1a

In the ﬁrst benchmark, we selected two schedules.

The ﬁrst schedule of a vehicle named 4 4a, which

serves 10 service strips. Mainly trips of lines 4 and

14, but the schedule is starting with a shortened ser-

vice strip of line 1 and once also servers line 5. The

second schedule 5

1a serves 10 service trips on line

5. Selected schedules are presented in Table 2. Indi-

vidual routes from selected schedules are illustrated

in Figure 4.

Table 2: Selected schedules for the benchmark 4 4a and

5 1a. The lines cowered by schedules are presented.

Vehicle schedule Line Stops count

4 4a

4 20

14 20

1m 13

5a 16

5b 15

5 1a

5a 16

5b 15

According to the optimal solution of the bench-

mark, 7477m of road segments were selected for

building overhead charging lines. The total length

of all road segments were 35566m, which means that

approximately 21% were selected. Selected road seg-

ments are listed in Table 3 and visually illustrated in

Figure 4.

Table 3: Selected road segments in optimal solution.

Starting node Ending node

Halkova Policia

Policia Hurbanova

Hurbanova Zeleznicna stanica

Sv. Cyrila a Metoda Obchodna

Jasenova Limbova

Polna Hlinska

Matice slovenskej Fatranska

Stefanikovo namestie Zeleznicna stanica

Hurbanova Policia

Mostna Hlinska

Polna Smrekova

Smrekova Limbova

Limbova Jasenova

According to benchmark results in Table 4, it is

clear, that adjustment of the model had a positive im-

pact on the process of solving the given problem. As

we can see the number of constraints increased, while

the number of variables stays the same. After the

pre-solve process, the number of variables decreased

signiﬁcantly speciﬁcally using edited model formula-

tion. Pre-solve in IP solver attempts to simplify the

problem by detecting and removing redundant con-

straints, tightening variable bounds, etc.

Moreover, using the original model formulation

the solver was unable to successfully ﬁnish the bench-

mark run, because of lack of available RAM. On the

other hand, the optimal solution was found using the

edited model. Running time was also shorter using

the edited model.

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

190

Figure 4: Illustration of bus lines 1, 4, 5 and 14 in Zilina.

Selected edges for installing overhead charging wires are

marked as red.

Table 4: Models comparation.

Original

model

Edited

model

Constraints (initial) 76 336

Constraints (pre-solved) 76 336

Variables (initial) 3816093 3816093

Variables (pre-solved) 2099867 1482018

Running time (s) 6134 3248

Objective

function value

7478 7477

Optimal solution

out of

memory

yes

6.2 Benchmark 24 5 and 26 3

In second benchmark we also selected two schedules

presented in Table 5.

The ﬁrst schedule of a vehicle named 24 5, which

serves 14 service strips. Mainly trips of line 24, but

also serves shortened service strip on line 30. The

second schedule 26 3 serves 11 service trips on line

26. Individual routes from selected schedules are il-

lustrated in Figure 5.

In this case, according to benchmark results in Ta-

ble 6, we can tell that the optimal solution was found

using both models. The optimization process was

Table 5: Selected vehicle schedules.

Vehicle schedule Line Stops count

25 5

24 20

24 7

30 6

26 3

26a 17

26b 18

26 9

Table 6: Models comparation.

Original

model

Edited

model

Constraints (initial) 108 380

Constraints (pre-solved) 107 379

Variables (initial) 4020976 4020976

Variables (pre-solved) 2295378 1106005

Running time (s) 800 1299

Objective

function value

10566 10566

Optimal solution yes yes

faster using the original model formulation.

The optimal solution of the second benchmark

tells that 10566m of road segments were selected for

building overhead charging lines. The total length of

all road segments were 53984m, which means that ap-

proximately 19.6% were selected. Selected road seg-

ments are listed in Table 7 and visually illustrated in

Figure 5.

Table 7: Selected road segments in optimal solution.

Starting node Ending node

Priehradna Furdekova

Hurbanova Zeleznicna stanica

Zeleznicna stanica Stefanikovo namestie

Predmestska Kosicka TESCO hyp.

Kosicka TESCO hyp. Pri celulozke

Potoky Dolna trnovska

Pri celulozke Kosicka TESCO hyp.

Predmestska Stefanikovo namestie

Stefanikovo namestie Zeleznicna stanica

Zeleznicna stanica Hurbanova

Kysucka Namestie hrdinov

Jastrabia Vrania

Vrania Jastrabia

Kysucka Zeleznicna stanica

Internatna Univerzitna

Internatna Vysokoskolakov plavaren

7 CONCLUSIONS

We can expect that the demand for the inclusion of as-

sisted trolleybuses technology in urban transport will

have an increasing trend. In this case, reliable ways of

solving related problems need to be researched (Gry-

gar et al., 2019).

This paper brieﬂy explains the required data con-

version approach and mathematical model required

for creating a minimal network of overhead wires that

Linear Model Adjustment and Approximate Approach for Creating Minimal Overhead Wires Network for Vehicle Schedules

191

Figure 5: Illustration of bus lines 24, 26 and 30 in Zilina.

Selected edges for installing overhead charging wires are

marked as red.

would be enough for deployment and operation of

such battery-assisted trolleybuses.

Performed computational tests demonstrate the

suboptimal approach on how to solve this problem

for selected vehicle schedules. According to the ﬁrst

benchmark, we can see, that performed condition ad-

justment in the model can be beneﬁcial for solving

similar problems. The presented solutions show that

around 20% coverage was needed for the operation

of such vehicles. We can see, that selected segments

were mostly in the centre of the city. This is caused

by the fact that most of the bus lines connect the city

centre with other areas.

We will try to improve our approach to future re-

search. Vehicle SOC needs to be preserved from the

end of a route to start of next one. After that, we will

be able to ﬁnd optimal solutions for vehicle schedules.

Lastly, the route’s nodes number can have a seri-

ous impact on the solvability of the problem. There-

fore, in future research, we will try to combine nearby

route segments or simplify the peripheral edges (Gry-

gar and Kohani, 2019b). We will also investigate

other methods of creating a more efﬁcient model.

Heuristic approach options may need to be considered

for large scale tasks as well.

ACKNOWLEDGEMENTS

The authors would like to thank VEGA 1/0689/19

”Optimal design and economically efﬁcient charging

infrastructure deployment for electric buses in pub-

lic transportation of smart cities” and APVV-15-0179

Reliability of emergency systems on infrastructure

with uncertain functionality of critical elements.

REFERENCES

Bartomiejczyk, M. (2017). Practical application of in mo-

tion charging: Trolleybuses service on bus lines. 2017

18th International Scientiﬁc Conference on Electric

Power Engineering (EPE), pages 1–6.

Bartomiejczyk, M., Stskala, V., Hrbac, R., and Poom, M.

(2013). Trolleybus with traction batteries for au-

tonomous running.

Bergk, F., Biemann, K., Lambrecht, U., Ptz, R., and

Landinger, H. (2016). Potential of in-motion charging

buses for the electriﬁcation of urban bus lines. Jour-

nal of Earth Sciences and Geotechnical Engineering,

6:347–362. Scienpress Ltd.

Ghlich, D., Kunith, A., and Ly, T. (2014). Technology as-

sessment of an electric urban bus system for berlin.

WIT Transactions on The Built Environment, 138:13.

20th International Conference on Urban Transport and

the Environment, Portugal.

Grygar, D. and Kohani, M. (2019a). Covering of the battery

assisted trolleybus ﬂeet schedules by minimal over-

head wires network. The 13th International Confer-

ence on Multiple Objective Programming and Goal

Programming (MOPGP), Marrakech, Morocco. In

print.

Grygar, D. and Kohani, M. (2019b). Data conversion and

exact approach to overhead wires network minimisa-

tion for the battery assisted trolleybus ﬂeet. The 15th

International Symposium on Operations Research in

Slovenia — 25th 27th September 2019, Bled, Slove-

nia.

Grygar, D., Kohani, M., Stefun, R., and Drgona, P. (2019).

Analysis of limiting factors of battery assisted trolley-

buses. Transportation Research Procedia, 40:229 –

235. TRANSCOM 2019 13th International Scientiﬁc

Conference on Sustainable, Modern and Safe Trans-

port.

He, F., Chen, Z., and Yin, Y. (2016). Optimal deployment

of charging lanes for electric vehicles in transportation

networks. Transportation Research Part B: Method-

ological, 91:344 – 365.

Hwang, I., Jang, Y., Ko, Y., and Lee, M. (2018). System

optimization for dynamic wireless charging electric

vehicles operating in a multiple-route environment.

IEEE Transactions on Intelligent Transportation Sys-

tems, 19(6):1709–1726. cited By 7.

Janacek, J. and Kvet, M. (2016). Sequential approximate

approach to the p-median problem. Computers & In-

dustrial Engineering, 94.

Janek, J., Kohni, M., Szendreyov, A., and ubo Buzna

(2015). Diskrtna optimalizcia. EDIS vydavatestvo

ilinskej univerzity. Fakulta riadenia a informatiky,

ilinsk univerzita, Katedra matematickch metd a oper-

anej analzy.

Janek, J. and Kovaikov, J. (1999). Porovnanie prstupov za-

loench na dulnom prstupe pre rieenie umiestovacch

loh. Komunikcie - Vedeck listy ilinskej univerzity,

pages 4–8. Vydanie: 2/99.

Janosikova, L., Jankovic, P., and Kvet, M. (2017). Improv-

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

192

ing emergency system using simulation and optimiza-

tion.

Janovec, M. and Kohni, M. (2019). Exact approach to

the electric bus ﬂeet scheduling. Transportation Re-

search Procedia, 40:1380 – 1387. TRANSCOM 2019

13th International Scientiﬁc Conference on Sustain-

able, Modern and Safe Transport.

Jaosikova, L. and Jankovic, P. (2018). Emergency medical

system design using kernel search. 2018 IEEE Work-

shop on Complexity in Engineering (COMPENG),

pages 1–6.

Kooten Niekerk, M., Akker, J., and Hoogeveen, J. (2017).

Scheduling electric vehicles. Public Transport, 9.

Kvet, M. (2019). Fast approximate algorithm for robust

emergency system design. pages 1–6.

Montoya, A., Guret, C., Mendoza, J. E., and Villegas, J. G.

(2017). The electric vehicle routing problem with

nonlinear charging function. Transportation Research

Part B: Methodological, 103:87 – 110. Green Urban

Transportation.

Pisko, M. (2017). Vyehradsk iniciatva pre transparentn

mestsk spolonosti. Prpadov tdia, Slovensko, Dopravn

podnik Bratislava, a.s., Transparency International

Slovensko.

Rice, D., Sderholm, T., and Mortensen, O. W. (2017).

Trolleybus extension to laksevg. Teknisk Forprosjekt.

Hordaland Fykeskommune.

Ritter, A., Elbert, P., and Onder, C. (2016). Energy saving

potential of a battery-assisted ﬂeet of trolley buses.

IFAC-PapersOnLine, 49(11):377 – 384. 8th IFAC

Symposium on Advances in Automotive Control AAC

2016.

Rogge, M., Wollny, S., and Sauer, D. U. (2015). Fast charg-

ing battery buses for the electriﬁcation of urban pub-

lic transporta feasibility study focusing on charging

infrastructure and energy storage requirements. Ener-

gies, 8(5):4587–4606.

Saft (2016). Sth ni-cd battery range - high power for hybrid

electric vehicles and automated guide vehicles. Saft

Industrial Battery Group. France.

Ushijima-Mwesigwa, H., Khan, M. Z., Chowdhury, M. A.,

and Safro, I. (2017). Optimal installation for electric

vehicle wireless charging lanes. Clemson University,

Clemson SC, USA. School of Computing, Department

of Civil Engineering.

Yang, S. C., Li, M., Lin, Y., and Tang, T. Q. (2014). Elec-

tric vehicle’s electricity consumption on a road with

different slope. Physica A Statistical Mechanics and

its Applications, 402:41–48.

Linear Model Adjustment and Approximate Approach for Creating Minimal Overhead Wires Network for Vehicle Schedules

193